Fresnel models for gravitational wave effects on pulsar timing

Merging supermassive black hole binaries produce low frequency gravitational waves which pulsar timing experiments are searching for. Much of the current theory is developed within the plane-wave formalism, and here we develop the more general Fresnel formalism. We show that Fresnel corrections to gravitational wave timing residual models allow novel new measurements to be made, such as direct measurements of the source distance from the timing residual phase and frequency, as well as direct measurements of chirp mass from a monochromatic source. Probing the Fresnel corrections in these models will require that future pulsar timing arrays include more distant pulsars across our galaxy, and greatly decrease the uncertainty on these pulsar distances (to within the order of the gravitational wavelength). However, we find that with these conditions met the measured source distance uncertainty can be made less than 10 per cent of the distance to the source for sources out to $\sim 100$ Mpc, source sky localization can be reduced to sub-arcminute precision, and source volume localization can be made to less than $1 \ \text{Mpc}^3$ for sources out to $1$ Gpc distances.